   Chapter 5, Problem 103RE ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
1 views

# Approximating the Area of a Plane Region In Exercises 103 and 104, use the rectangles to approximate the area of the region. Compare your result with the exact area obtained using a definite integral. f ( x ) = x 3 ,     [ 0 , 3 ] To determine

To calculate: The area of the region bounded by the graph of the function f(x)=x3 in the interval [0,3] with the help of midpoint rule for rectangle to approximation and compare the result with exact area.

Explanation

Given information:

The functions, f(x)=x3 and the provided intervals is [0,3]. The provided figure is shown below:

Formula used:

Steps to solve a definite integral abf(x)dx with the help of midpoint rule.

Step 1: For a given interval [a,b] divide it into n subintervals with having width of,

Δx=ban

Step 2: Evaluate the midpoint for the given subinterval. Midpoints={x1,x2,x3,xn}

Step 3: Find the value of f at each midpoint and make the sum as shown below:

abf(x)dxban[f(x1)+f(x2)+f(x3)++f(xn)]

Fundamental theorem of calculus:

abf(x)dx=F(x)]ab=F(b)F(a)

Calculation:

Consider the provided function,

f(x)=x3

The graph is divided into 3 rectangles from 0 to 3 therefore. The intervals are [0,3] with n=3

Now divide the provided interval into 3 subparts as shown below,

Δx=303=33=1

Therefore the 4 subintervals are [0,1],[1,2] and [2,3].

Now find the mid points of these intervals because each subinterval has a width of 1. Therefore, the mid points of these intervals are 12,32 and 52.

Here, the mid points 12,32 and 52 lies in the middle of [0,1],[1,2],[2,3] intervals respectively

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

#### In problems 5-8, write the following sets in a second way. 6.

Mathematical Applications for the Management, Life, and Social Sciences

#### Convert 72FtoC.

Elementary Technical Mathematics

#### True or False: The graph at the right is the graph of an even function.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 